A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

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Integrating probabilities

My following problem is of general nature, here is an example to illustrate it. For example let $\left(\xi_i\right)_{i \geq 1}$ be independent and identically Exp(1) distributed random variables. We ...
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Advice on Stochastic Drift Sequence

I recently wrote a paper for university reviewing stochastic differential equations and it pretty well. I now have an idea for a paper which I would like to write in my spare time (and maybe if I can ...
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Jeffrey's Prior for Bivariate Lognormal

Exactly what the question says, I'm working on code for an MCMC simulation and need to set some uninformative or weakly informative priors. I haven't been able to find the prior for the sigma ...
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21 views

Poisson Process Suitable Scenarios

I have a couple of doubts about if these scenarios are suitable to be modeled as a Poisson process. I will like to have your views and arguments why. Packets are lost due to packet overflow in the ...
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61 views

A Markov Chain Problem.(Change the color of ball)

There are $n$ different color balls in a box. Take two balls in turns, and change color of the second ball to the first. (This is one operation). Let $k$ be the (random) number of operations needed to ...
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Changing a queueing processes

I am wondering if there are any general results related to how queue behavior changes if one is allowed to repeatedly make one-off behavior changes. Situation Consider a general queueing system ...
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Prove this is markov chain [on hold]

Let $Y_i = i$ with probability $p_i$. Let $X_i = max[Y_i , Y_i-1, ...]. $ We have $i={0,1,2,3,4,...}. $ Prove that $X_i$ is a Markov chain and write down its matrix.
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29 views

Poisson process has independent and stationary increments

Being $N_t$ a Poisson process, defined as $$N_t:=\sum_{n\geq 1} n \mathbb{1}_{[T_n,T_{n+1}[}(t)$$ where $T_n$ are sums of independent exponential random variables, how can I prove it has stationary ...
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23 views

Will the branching process go extinct with probability 1?

I am trying check whether the branching process goes extinct with probability one. Single Type Branching Process with Pk = (1/2n)(n/k), for k = 0,.....,n with n > 2. Assuming, i can be able to ...
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37 views

How to apply the strong Markov property in this case?

I'm trying to understand the following proof: Theorem: Let $(X_n)$ be an irreducible $(\alpha, \mathbf p)$-Markov chain with a finite state space $S$. Then $(X_n)$ is positive recurrent. ...
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How to get closed form solutions to stopped martingale problems?

Way back when, I took a course in stochastic processes in college. I remember being frustrated by the plethora of abstract proofs without much in the way of how to use them to get actual results. It ...
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Explicit solution SDE?

I have the following SDE: $$dY_{t}=A\left(\frac{W_{t}^{1}}{\sqrt{t}},\frac{Y_{t}}{\sqrt{t}}\right)dW_{t}^{1}+B\left(\frac{W_{t}^{1}}{\sqrt{t}},\frac{Y_{t}}{\sqrt{t}}\right)dW_{t}^{2}$$ where ...
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For a Brownian motion prove that (a) $N (t) -λt $ and (b) $e^{(\log(1-u) N (t) + uλt)}$, are martingales

For a Brownian motion ${z (t)}$ and for any $β ∈ R$, be $V (t) = \exp\{ βz (t) - (t β ^ 2) / 2\}, t≥0 $ Show that ${V (t)}$ is a martingale with respect to a Brownian filtration. Also ${N (t)}$ be a ...
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47 views

Mean and variance of a stochastic process

Let \begin{equation} \begin{array}{l} y_1(t)=e^{-\kappa_1 t}y_1(0)+\displaystyle\int_0^t\kappa_1 e^{\kappa_1(s-t)}\theta_1ds +\sigma_1\displaystyle\int_0^te^{\kappa_1(s-t)}\sqrt{y_1(s)}dZ_1(s),\\ ...
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Is $W_{2t}-W_t$ a brownian motion?

Is $W_{2t}-W_t$ a brownian motion? $(W_t)_{t\geq 0}$ is a brownian motion, I have to show that $X_t:=W_{2t}-W_t$ is a brownian motion as well. $$W_{2t}= 1/\sqrt{2} W_t$$ (by scaling property) then ...
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Centered independent increments process is a martingale

Let $(X_n)$ be an centered integrable process with independent increments (which as far as I understand means that $(X_{n+1}-X_n)_{n\in \mathbb N}$ is independent). While showing that $(X_n)$ is a ...
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$E[M_t|H_t]$ is a martingale with respect to $H=(H_t)_{t\geq 0}$, $H_t \subset \mathcal{F}_t \forall t$

Being $(M_t)_{t \geq 0}$ an $\mathcal{F}$-martingale, I have to show that $E[M_t|H_t]$ is a martingale with respect to $H$ ($H=(H_t)_{t\geq 0}$, $H_t \subset \mathcal{F}_t \forall t$). I proceded ...
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38 views

Write down the HJB equation

Suppose that we have to solve the following optimal control problem \begin{align} V(t,x) = \min_{\alpha}\mathbb{E} \left[\int_{0}^{T}L(t,x,\alpha)dt + F(e^{-\beta t}X^{\alpha}_{T})\right] ...
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Do we need $\tau \leq \nu$ to show $E(X_\tau)=E(X_\nu)$?

My lecture notes claim that if $(X_n)$ is a martingale and $\tau$ is a stopping time bounded by $N$ then $$E(X_\tau)=E(X_{\tau \wedge N})=E(X_{\tau \wedge 0})=E(X_0)$$ and then remarks that if $\tau$ ...
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Random number distribution from a different distribution

Suppose I have a random number generator that generates random numbers $x$ with a normal distribution $p(x) \propto e^{-x^2}$ (modulo normalization, but lets keep it simple). Now, out of these ...
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Proof that a random variable has exponential distribution.

Supose that $X_1$ is a continuous and positive (real) random variable with exponential distribution, namely $$P(X_1>t)=e^{-\lambda t}\quad t>0$$ Now suppose that $X_2$ is another continuous and ...
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Find one-dimensional distribution function $F(y\mid t)$ of random process $Y(t)$

$ Y(t)=tZ^2;\quad Z\sim U(-2;2); \quad t\ge0. \quad$ I need to 1) find one-dimensional distribution function $F(y|t)$ of random process $Y(t)$. 2) calculate probability that trajectory of the ...
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characteristic function of a stochastic process with stationary and independent increments

Let $(X_t)_{t\geq 0}$ be a stochastic process with independent and stationary increments. I have to show that $E[e^{itX_1}]=\phi^n(t)$ Since increments are independent, I can write ...
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32 views

Stochastic processes with full memory

Markov processes are stochastic processes with no memory. How are called the stochastic processes with full memory? Can't found anything on google.
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How to calculate radius density of observation using locality sensitive hashing?

How do I calculate radius density of observation using locality sensitive hashing? I am new to the locality sensitive hashing(LSH). LSH based learning and Querying was difficult to understand.
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18 views

Max of independent and identical random variables is Markov

I'm supposed to show that given a sequence $\{Y_n\}$ of i.i.d the stochastic process $$X_n=\max(Y_0, Y_1...,Y_n)$$ is a Markov of chain. I think I could do it by induction but I would rather see how ...
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57 views

Angle bracket and sharp bracket for discontinuous processes

The question is quite simple actually. I am trying to understand the differences between the angle bracket $\left<X,Y\right>$ of two processes with jumps $X,Y$, and the sharp bracket of $[X,Y]$. ...
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38 views

Stopping times problem: $ \tau_+ = \inf \{t \ge 0 \mid W_t>0\}$

Stopping times problem, $\tau_+ = \inf \{t \ge 0 \mid W_t>0\}$ I can not prove the following : P/S: When I look at the stopping time, I feel that $\{W_0 > 0\} = \{\tau_+ = 0\}$ , is that ...
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Expected value and variance of random process

Let $U,V$ be random variables with distributions $\mathcal{U}(-1,1)$ ,$\mathcal{E}(2)$ (uniform and exponential). If $U$ and $V$ are independent what is the variance and expectation of the random ...
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If a random integral has moments of all orders, is the same true for its kernel?

Suppose you have a continuous semimartingale $S_t=M_t + A_t$ where $A_t$ is the continuous finite variation part which has the form $A_t = \int_0^t b_s \, \mathrm{d} s$, where $\int_0^{\infty} |b_s| ...
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Stopped supremum of the Brownian local time still $L^p$ bounded in space?

Let $B_t$ be a standard Brownian motion and $L_t^x$ its local time in $x$ at time $t$. For fixed $t$ and $p>1$, it holds that $$ \sup_{x \in \mathbb{R}} \operatorname{E} [ (L_t^x)^p ] < ...
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Conversion of continuous, linear stochastic system to discrete, LQR/LQG

I have the standard stochastic, linear time varying system $dx(t) = (A(t)x(t) + B(t)u(t))dt + G(t)dw(t) $ with $x(t_0) = x_0$ with quadratic cost $J = x(t_F)^TQ_Fx(t_F) + \int_{t_0}^{t_F}\left( ...
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kullback liebler divergence for correlated processes

Suppose $X_n^{(1)}=\lambda_1 X_{n-1}^{(1)}+\mu_1+\epsilon_n^{(1)}$ and $X_{n}^{(2)}=\lambda_2X_{n-1}^{(2)}+\mu_2+\epsilon_n^{(2)}$ where $|\lambda_i|<1$ for $i=1,2$ and $\epsilon_n^{(i)}$ are ...
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A Poisson process question

I saw an old post here, claiming that for a Poisson Process $X(t)$: $P[X(t) - X(s) = 1 \mid X(t) = 4]=\frac{4(t-s) s^3}{t^4}$ Am I missing something essential about stochastic processes, probability ...
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A question about a Markov Chain

I encountered a question about Markov Chains which looks interesting. Given a homogeneous, irreducible, non cyclic Markov Chain with $K$ possible states and a transition matrix $Q$. We define $T_i$ ...
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Comparing hitting time of two random walks

There are two random walks, $S^t_i=S^{t-1}_i+ X_i^t$ for $i=1,2$, $X^t_i$ i.i.d they have boundaries $h_1$ and $h_2$ respectively. I'm wondering if it's possible to calculate the probability that one ...
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Mean value theorem inside the Expectation

Consider a stochastic process $X_t$ with continuous paths. I'd like to apply the mean value theorem inside the expectation, i.e. write something like $$ \operatorname{E} \left[ \int_0^t X_s \, ...
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63 views

Derivation of Black-Scholes equation by riskless portfolio

The following is a summary of the derivation of the Black-Scholes equation as given on wikipedia (http://en.wikipedia.org/wiki/Black-Scholes_equation#Derivation) - I have a question regarding the ...
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Approximating the probability of an event by finite-dimensional distributions

Let $(X(t))_{t\ge 0}$ be a stochastic process on $\mathbb{R}^d$, say an Ito diffusion (with continuous sample paths). Let $A\subset \mathbb{R}^d$ be a measurable set and $t>0$. Does the following ...
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Markov property for a stochastic process with discrete state space.

Consider a stochastic process $\{X_s\}_{s\in\mathcal S\subseteq\mathbb R}$ with value in $(\mathbb R,\mathcal B(\mathbb R))$ adapted to a filtration $\{\mathcal F_s\}$ (we can suppose that ...
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Filtration and measure change

I'm reading Steven E. Shreve's "Stochastic calculus for finance II", and find myself not really understand the concept of "filtration". Yes, the definition of filtration is straight forward, it's ...
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Find the probability generating function of a GW process

Consider a Galton-Watson process with offspring distribution $\mathrm{Poisson}(1)$. That is, $\textbf{p}(k) = \frac{e^{-1}}{k!}$. Given this information, and that $P(z) = ...
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Distribution of number of Poisson arrivals in interval

$X_1$ and $X_2$ are both Poisson processes. $N$ is the number of arrivals of $X_1$ in between two subsequent arrivals of $X_2$. Derive the probability density $f_N(n)$ of $N$. I wanted to start from ...
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Distribution of maximum/minimum proportion in a sampling process

I am facing something that can be explained as a balls & urns problem. Suppose you have $B$ black and $W$ white balls inside an urn. They are randomly chosen, one by one, without replacement, and ...
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Proof of finite expectation of renewal process (2) [duplicate]

I don't know if it is allowed here, to repost again his own question. I hope it is ok... I already asked this question here: Finite expectation of renewal process But I don't understand the last steps ...
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Supply the transition matrix for these (possible) Markov chains

Reading Grimmet, Stirzaker: Probability and Random Processes, which unfortunately doesn't have solutions. Trying to make sure I understand Markov chains. A die is rolled repeatedly. Which of these ...
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Covariance of a function of random variables

I want to find the covariance $K_X(t,t')$ of the following signal $X(t)$: $X(t)=\sum\limits_{n=-\infty}^{+\infty} A_np(t-nT)$ where $ p(t) = \begin{cases} \ 1 & \text{if } 0<t\leq T/2 ...
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The meaning of the connection between power spectral density and auto correlation

I know that if we have a signal $x(t)$, then its Fourier transform would be the signal in the frequency space, which I understand to be how much of each frequency exists in the x(t) signal. $ ...
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In stochastic calculus, why do we have $(dt)^2=0$ and other results?

I'm doing actuarial problems of Exam MFE and it covers some of the stochastic calculus (like Ito's Lemma). One of the frequently used results are the so-called "multiplication rules": $(dt)^2=0$ ...
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Multi-dimensional Feynman Kac Theorem

I am trying to understand how to prove the multi-dimensional version of the Feynman-Kac formula. The single-dimensional version is proved on this page: en.wikipedia.org/wiki/Feynman–Kac_formula ...